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Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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21356 questions
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Algebraic Topology Hatcher Chapter 3.1 Problem 13

The Problem: Let $\langle X, Y\rangle$ denote the set of basepoint-preserving homotopy classes of basepoint preserving maps $X\rightarrow Y$. Using Proposition 1B.9, show that if $X$ is a connected CW complex and $G$ is an abelian group, then the…
Math_Day
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Obtaining a Quotient Space of $\mathbb{R}^2$

Using stereographic projection of a sphere, $S^2$, we can obtain the one point compactification of $\mathbb{R}^2$ is sphere, i.e. $S^2$ can be thought of as $\mathbb{R}^2 \cup \{ \infty \}$. Now I am wondering how can $S^2$ be obtained by…
SJA
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About quotient topology

I am studying the S G Krantz's book, where in the quotient topologies section, while defining the "the space obtained from $X$ by collapsing the subset $E$ to a point"(it is nothing but, taking a subset of the original topological space and obtain a…
SJA
  • 174
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How does null-homotopic loops exist in torus?

If we identify torus by $S^{1} \times S^{1}$, then what are the ways null-homotopic loops exist in torus? I don't see a way to do this except we walk around the genrator back and forth. Since if we consider two genrators a,b for the torus, then any…
Sophie
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Quotient space from the 8-sided polygonal region; Munkres

Let $X$ be the quotient space obtained from an 8-sided polygonal region $P$ by pasting its edges together according to the labelling scheme $acadbcd^{-1}d$. a) Check that all vertices of $P$ are mapped to the same point of the quotient space $X$ by…
TJIF
  • 463
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Proving a map is not a chain homotopy equivalence.

Let $A$ be a point on a circle in $\mathbb{R}^2$ and let $B$ be the complement of $A$ in this circle. How do I show that $\{A,B\}$ is not an excisive pair? I say wlog, let $U=\{z\in \mathbb{R}^2:|z|=1\}$, $A=\{(1,0)\}$, and $B=U-A$. Then we claim…
Johnathon Taylor
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Fundamental group of the torus minus a point (Van Kampen thm)

i had the exercise to compute the fundamental group of the torus minus one point p. I know that the fundamental group of the torus is $\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1) = \Bbb Z \times \Bbb Z$. So: $U :=$ open neighborhood of p $V := T^2…
Ton910
  • 145
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Simpler proof of van Kampen's theorem?

I've been trying to understand the proof of van Kampen's theorem in Hatcher's Algebraic Topology, and I'm a bit confused why it's so long and complicated. Intuitively, the theorem seems obvious to me. Given a path $p$ in $A \cup B$, we can split it…
user806826
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Why is it that no classical surface, except the sphere, is a suspension over any space?

In the book 'Homotopical Topology', exercise 4, chapter 2 asks: Prove that no closed (that is, without holes) classical surface except $S^2$ is homeomorphic to a suspension over any other space. It looks like, we have to show any genus g surface…
Isomorphism
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How can I construct a homotopy between a constant function to a continuous function?

I have a few questions in mind and I would really appreciate if I can clear some questions bogging my mind. Let $X$ be a topological space. Is there any loop based at $x_0$ that is not homotopic to a constant function based at $x_0$? Is every…
James C
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Van Kampen with complicated attaching map

I would like to know if it is possible to use van Kampen's theorem without knowing exactly what the intersecting space looks like. So given $U$ and $V$, and an attaching map, is it possible to work out the fundamental group of $X=U\cup_{f}V$…
BlackAdder
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Describe two non-homeomorphic connected irregular covering spaces of $S^1\vee S^1$

I am having trouble with the following qual problem. If someone could help me get started it would be great. Describe two non-homeomorphic connected irregular covering spaces of $S^1\vee S^1$, each of degree 3. Explain why your examples are…
Caroline
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Entangled circle in a solid torus (follow up)

I asked this question here. Can someone tell me if this is right: claim: There are no retractions $r:X \rightarrow A$ proof: (by contradiction) (i) If $f:X \rightarrow Y$ is a homotopy equivalence then the induced homomorphism $f_* : \pi_1(X, x_0)…
Rudy the Reindeer
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Show that the index of $A$ in $\pi_1(Y, f(x))$ is finite and divides $m$.

I am studying for a qualifying exam problem and I am having difficulties with this problem. I feel that the solution involves lifting properties, but I am not quite sure how to start. Let $X$ and $Y$ be closed connected oriented 2ñmanifolds, and…
Kristie
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Is there a topological space $X$ whose fundamental group depends on the base point?

Is there a space $X$ such that $\pi_1(X, x) \ne \pi_1(X, y)$ where $x \ne y$? I have not been able to think of an example. Part of the reason is that I have not computed too many fundamental groups yet.
user73920
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